Abstract In this paper, we define a quaternionic operator whose scalar part is a real parameter and vector part is a curve in three dimensional real vector space R 3 . We prove that quaternion product of this operator and a spherical curve represents a ruled surface in R 3 if the vector part of the quaternionic operator is perpendicular to the position vector of the spherical curve. We express this surface as a 2-parameter homothetic motion using the matrix representation of the operator. Furthermore, we define another quaternionic operator and show that each ruled surface in R 3 can be obtained by this operator. Finally, we give the geometric interpretations of these operators with some examples.

中文翻译：

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由四元数构造的直纹曲面

摘要 本文定义了一个四元数算子，其标量部分为实参数，向量部分为三维实向量空间R 3 中的曲线。我们证明，如果四元数算子的向量部分垂直于球面曲线的位置向量，则该算子与球面曲线的四元数积表示 R 3 中的规则曲面。我们使用算子的矩阵表示将此表面表示为 2 参数的同位运动。此外，我们定义了另一个四元数算符，并表明 R 3 中的每个规则曲面都可以通过该算符获得。最后，我们通过一些例子给出了这些算子的几何解释。